## Thursday, October 27, 2011

### Trigonometry

When we talks about trigonometry, we will most probably think of trigonometric functions, e.g. sine, cosine, tangent etc. Ever think of how were the graphs of these functions obtained?

Let's consider a right triangle with hypotenus of 1 unit. You may drag the point C below and observe the change of the angle. Recall that

$\sin\theta=\frac{y}{r}$
$\cos\theta=\frac{x}{r}$
$\tan\theta=\frac{y}{x}$

Point E is (x, sin x), point F is (x, cos x) and point G is (x, tan x). Observe the position of point E, F or G as you drag point C. You may choose the desire graph at the right bottom corner of the graph.

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If you wish to know more about trigonometric functions, please click on the function you are interested under Trigonometry at the left column.

Betty, Created with GeoGebra

## Friday, October 21, 2011

### Hyperbolic Tangent & Hyperbolic Cotangent

Hyperbolic tangent is defined as

$\fn_phv \tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$,

while hyperbolic cotangent is the reciprocal of hyperbolic tangent, and it is defined as

$\fn_phv \coth(x)=\frac{\cosh(x)}{\sinh(x)}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$.

If we want to sketch the graph of the earlier function, think of the y-intercept and limits below:

If we want to sketch the graph of the later function, think of the limits below:

The concept above applied for all values of a > 0. You may slide the value of 'a' below to look at the changes of $\fn_phv \tanh(ax)$ and $\fn_phv \coth(ax)$ for different values of 'a'.

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Betty, Created with GeoGebra

### Hyperbolic Cosine & Hyperbolic Secant

The hyperbolic cosine function is defined as
$\cosh(x)=\frac{e^{x} +e^{-x}}{2}$

or generally,
$\cosh(ax)=\frac{e^{ax} +e^{-ax}}{2}$.

Look at the graph below, try to slide the value of 'a' to different values to view the changes on graph.

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Now, take a look at the graph of the reciprocal of hyperbolic cosine function, i.e. hyperbolic secant function, which is defined as
$\textup{sech}(ax)=\frac{1}{\cosh(x)}=\frac{2}{e^{ax} +e^{-ax}}$

Observe that when a = 1,

Again, slide the value of 'a' to view the changes of the graph.

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Betty, Created with GeoGebra

### Hyperbolic Sine & Hyperbolic Cosecant

The hyperbolic sine function is defined as
$\sinh (x)=\frac{e^x -e^{-x}}{2}$

or generally,
$\sinh (ax)=\frac{e^{ax} -e^{-ax}}{2}$

Look at the graph below, try to slide the value of 'a' to different values to view the changes on graph.

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Now, take a look at the graph of the reciprocal of hyperbolic sine function, i.e. hyperbolic cosecant function, which is defined as
$\textup{csch} \; (ax)=\frac{1}{\sinh{ax}}=\frac{2}{e^{ax} -e^{-ax}}$.

Observe that when a =1,
Again, slide the value of 'a' to view the changes of the graph.

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Betty, Created with GeoGebra

## Monday, September 19, 2011

### Trapezoidal Sum

When upper sum and lower sum give a range of estimation to the area below a curve y = f(x) for a < x < b, Trapezoidal Sum uses n numbers of trapezoids to do the estimation of the area.

Look at the graph below, let's consider when n = 2 (2 trapezoids are built). Do take note that for each trapezoid, two points are taken from the left-ended value and right-ended value of the subinterval as part of the vertives. The summation of area of all the trapezoids, 12.47 square units.

If you slide the value of n to higher value, you may see that the trapezoids are narrowed down and the estimation of area below the curve is getting nearer to the actual area below the curve.

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Betty, Created with GeoGebra

## Sunday, July 10, 2011

### Boxplot

Let say we have a set of data and we would like to display the following statistics:
1) Mean
2) Mode/Modes
3) Median
4) First Quartile
5) Third Quartile
6) Maximum
7) Minimum

Item 3) - 7) can be viewed using boxplot (or sometimes also called as box-and-whisker plot). Observe the boxplot below, data are keyed into cells of A1 to A25. You may change the values in the stated cells and see the changes of the statistics and the boxplot as well. However, the position of the statistics won't change even if there is a change on the boxplot. Please take note the meaning of each vertical line of the boxplot before you start to change the values in the stated cells.

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Betty, Created with GeoGebra

## Thursday, June 30, 2011

### Parametric

A pair or a set of parametric equations may look like
$x =2,y=t, \; t\; \in\; (-\infty,\infty)$,
or
$x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi]$,
or
$x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]$,
or
$x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty)$.

See the common feature?

Now, let's try to combine the variables:
a) $x =2,y=t, \; t\; \in\; (-\infty,\infty)\; \; \Leftrightarrow \; \; x=2$

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This is a vertical line x=2.

b) $x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi] \Leftrightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$

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This is a vertical ellipse centered at the origin with vertices at (0,3) and (0,-3), and endpoints of minor axis at (2,0) and (-2,0).

c) $x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]\Leftrightarrow y=2\sin x$

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This is a sine function with magnitude 2.

d) $x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty) \Leftrightarrow z^2=xy$
This is an object in 3-dimension.

You see, parametric equations can form a lot of curve with different type:
a) function, non-function;
b) line, curve;
c) 2-D, 3-D;
etc.

Can you try to build one set of parametric equations and figure out the graph of the set of equations?